Appears in collection : Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

In joint work with Mavraki a few years ago, we studied the arithmetic intersection numbers of sections of elliptic surfaces, defined over number fields. One consequence was a Bogomolov-type extension (and new proof) of a theorem of Barroero and Capuano, addressing a case of the Zilber Pink conjectures. I will describe the underlying geometric features of this problem and formulate related open questions about families of maps (dynamical systems) on $P^1$.